Visualizing Systems of Equations

GeoGebra

Ana EscuderFlorida Atlantic University

aescuder@fau.eduDuke Chinn

Broward County Public Schoolsjames.chinn@browardschools.com

GeoGebra

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System of Linear EquationsAn 8-pound mixture of M&M’s and raisins costs $18. If a lb. of M&M’s costs $3, and a lb. of raisins costs $2, then how many pounds of each type are in the mixture?

€

x + y = 83x + 2y =18 ⎧ ⎨ ⎩

x lbs of M&M’s

Y lbs of raisins

Gauss’ Method of Elimination

If a linear system is changed to another by one of these operations: (1)Swapping – an equation is swapped with

another(2)Rescaling - an equation has both sides

multiplied by a nonzero constant(3)Row combination - an equation is replaced by

the sum of itself and a multiple of another

then the two systems have the same set of solutions.

Restrictions to the Method• Multiplying a row by 0

that can change the solution set of the system.

• Adding a multiple of a row to itselfadding −1 times the row to itself has the

effect of multiplying the row by 0. • Swapping a row with itself

it’s pointless.

Example

€

x + y = 83x + 2y =18 ⎧ ⎨ ⎩

Multiply the first row by -3 and add to the second row. Write the result as the new second row

€

−3x − 3y = −24 3x + 2y =18 − y = −6

€

x + y = 8−y = −6 ⎧ ⎨ ⎩

Example (Cont)Add the two rows to eliminate the y in the first row

Write the result as the new first row€

x + y = 8−y = −6 ⎧ ⎨ ⎩

€

x = 2−y = −6 ⎧ ⎨ ⎩

Multiply the second row by -1

€

x = 2y = 6 ⎧ ⎨ ⎩

€

x + y = 83x + 2y =18 ⎧ ⎨ ⎩

Changed to

Geometric Interpretation

Possible Types of Solutions

Uniquesolution

No solution

Infinite solutions

General Behavior of Linear Combination

• If solution exists - the new line (row combination) passes through the point of intersection (solution).

• If no solution – the new line is parallel to the other lines

• If infinite solutions – the new line overlaps the other two.

In General…

€

ax + by = cdx + ey = f ⎧ ⎨ ⎩

€

a b cd e f ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Unique solution if:

€

ae −bd ≠ 0

Assuming a, b, c, d are not equal to 0

3 x 3 System of Equations

€

A1x + B1y +C1z = D1

A2x + B2y +C2z = D2

A3x + B3y +C3z = D3

⎧ ⎨ ⎪

⎩ ⎪

€

A1 B1 C1 D1

A2 B2 C2 D2

A3 B3 C3 D3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Possibilities with Systems of equations in 3 Variables

• Unique solution – A point

Infinite Solutions

No solution

Solving a System of Equations

€

1x +1y + 2z = 8−1x − 2y + 3z =13x − 7y + 4z =10

⎧ ⎨ ⎪

⎩ ⎪

€

1 1 2 8−1 − 2 3 1 3 − 7 4 10

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R1 + R2Replace row 2

€

1 1 2 8 0 −1 5 9 3 − 7 4 10

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

The new plane (blue) is parallel to the x-axis

-y + 5z = 9

€

1 1 2 8 0 −1 5 9 3 − 7 4 10

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R1 + R2Replace row 1

€

1 0 7 17 0 −1 5 9 3 − 7 4 10

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

The new plane (red) is parallel to the y-axis

€

1 0 7 17 0 −1 5 9 0 7 17 41

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Green plane is now parallel to the x-axis

€

1 0 7 17 0 −1 5 9 3 − 7 4 10

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R1*3 – R3Replace row 3

€

1 0 7 17 0 −1 5 9 0 7 17 41

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

(R2*7 + R3)/52Replace row 3

Green Plane is perpendicular to the z-axis

€

1 0 7 17 0 −1 5 9 0 0 1 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

1 0 7 17 0 1 0 1 0 0 1 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Blue plane is perpendicular to the y-axis

€

1 0 7 17 0 −1 5 9 0 0 1 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R3*5 – R2Replace row 2

€

1 0 7 17 0 1 0 1 0 0 1 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R1 - R3*7 Replace row 1

Red plane perpendicular to the x-axis

€

1 0 0 3 0 1 0 1 0 0 1 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

New Equivalent System

€

1x +1y + 2z = 8−1x − 2y + 3z =13x − 7y + 4z =10

⎧ ⎨ ⎪

⎩ ⎪ To

€

1x = 31y =11z = 2

⎧ ⎨ ⎪

⎩ ⎪

Eight Possibilities

What is the solution?

€

2x + 3y − 4z = −115x + 5y + 5z = 6−6x − 9y +12z = −14

⎧ ⎨ ⎪

⎩ ⎪

Two parallel planes intersected by a third plane

What is the solution?

€

−2x + 3y + 5z = 24x − 6y −10z = 8x −1.5y − 2.5z = −3

⎧ ⎨ ⎪

⎩ ⎪Three parallel planes

What is the solution?

€

3x + 2y − z =10x + 4y + 2z = 34x − 24y − 20z = 4

⎧ ⎨ ⎪

⎩ ⎪

Eliminate the same variable from at least

two equations

What is the solution?

€

3x + 2y − z =10x + 4y + 2z = 34x − 24y − 20z = 4

⎧ ⎨ ⎪

⎩ ⎪

Three non-parallel planes that form a

type of triangle

CAS in GeoGebra 4.2

Information• Downloading GeoGebra 5.0• Construction of ggb files

Uploaded in the Conference Online Planner and Conference App

• Power point and ggb fileshttps://floridageogebra.wikispaces.com/Conferences

Thank You!Ana Escuder

aescuder@fau.edu

Duke Chinnjames.chinn@browardschools.com

Special thanks to:Barbara Perez